NASA Technical Reports Server (NTRS) 20070010747: Huygens Titan Probe Trajectory Reconstruction Using Traditional Methods and the Program to Optimize Simulated Trajectories II

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by  NASA Technical Reports Server (NTRS)

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Huygens Titan Probe Trajectory Reconstruction Using Traditional Methods and the
Program to Optimize Simulated Trajectories II

Scott A. Striepe, NASA Langley Research Center, Hampton, VA
Robert C. Blanchard, National Institute for Aerospace, Hampton, VA
Michael F. Kirsch, National Institute for Aerospace, Hampton, VA
Wallace T. Fowler, The University of Texas at Austin, Austin, TX

Abstract
On January 14, 2005, ESA’s Huygens probe separated from NASA’s Cassini
spacecraft, entered the Titan atmosphere and landed on its surface. As part of
NASA Engineering Safety Center Independent Technical Assessment of the
Huygens entry, descent, and landing, and an agreement with ESA, NASA
provided results of all EDL analyses and associated findings to the Huygens
project team prior to probe entry. In return, NASA was provided the flight data
from the probe so that trajectory reconstruction could be done and simulation
models assessed. Trajectory reconstruction of the Huygens entry probe at Titan
was accomplished using two independent approaches: a traditional method and
a POST2-based method. Results from both approaches are discussed in this
paper.

INTRODUCTION

On January 14, 2005, the European Space Agency’s (ESA) Huygens probe
separated from NASA’s Cassini spacecraft, entered the Titan atmosphere and landed on
its surface.1-3 This probe used a multiple parachute system to enable atmospheric
measurements to be recorded during the probe’s over two hour descent to the surface.
Digital images, radar altimetry, accelerometer data, and Earth-based radio telescope
observations were also gathered during the entry, descent, and landing (EDL).4-9 Figure 1
illustrates the Huygens probe’s EDL profile. After atmospheric interface at 1270 km
above the surface, the probe decelerates to around Mach 1.5 at pilot parachute deploy.
This deploy event (designated T0) is triggered by a sequence of time and acceleration
conditions and is the epoch time for all subsequent events and most data sets generated
during the parachute descent phase. Three parachutes were used in the Huygens probe
system: (1) a 2.5-sec Pilot parachute; (2) a 15-minute Main parachute; and (3) a 2.5-hour
Drogue parachute. Data taken during the descent was relayed through the Cassini
spacecraft as it flew by Titan.

AAS/AIAA Space Flight Mechanics Meeting, Sedona, AZ, 28 Jan – 1 Feb 2007
1
Figure 1. Huygens Titan probe Entry, Descent, and Landing Sequence2

NASA Langley was involved in the pre-entry EDL analyses of the Huygens
probe. Analyses were conducted under the auspices of the NASA Engineering Safety
Center’s (NESC) Independent Technical Assessment (ITA) of the Cassini/Huygens probe
EDL at Titan.10 A Program to Optimize Simulated Trajectories II11 (POST2)-based
trajectory simulation was developed that included models of the probe aerodynamics,
parachute trigger logic and drag models for the Pilot, Main, and Drogue parachutes. As
part of the agreement with ESA, NASA provided results of all analyses and presented
findings to both the Cassini and Huygens project teams. In return, NASA was provided
the flight data from the probe so that trajectory reconstruction could be done and
simulation models assessed. NASA has completed similar assessments of flight data to
improve simulation models (such as capsule aerodynamics, parachute aerodynamics, and
atmospheric density and winds) for the Mars Pathfinder EDL, Mars Odyssey
aerobraking, and Mars Exploration Rovers EDL.12-14 ESA had a team that provided the
official project reconstruction of the Huygens EDL trajectory (the Huygens Descent
Trajectory Working Group, or DTWG).3 The main objective of this NESC sponsored
activity was to reconstruct the Huygens Probe data to improve NASA’s aerodynamics,
atmospheric density and winds, and parachute performance models. The NASA NESC
ITA team also analyzed the flight data to provide an independent trajectory
reconstruction. The results of these analyses were provided to the DTWG and the
Huygens Data Analysis Working Group (DAWG), with interaction between NASA,
DTWG, and DAWG to discuss any differences in the reconstructed trajectory.
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BACKGROUND

Trajectory reconstruction of the Huygens entry probe at Titan was accomplished
using two independent approaches. One was a traditional approach to trajectory
reconstruction that integrates the accelerometer measurements directly during the entry
phase (forward) and integrates the hydrostatic equation using the temperature and
pressure measurements to determine altitude and velocity (backwards). The other
approach used a Kalman filter module developed for reconstruction in conjunction with
the POST2-based simulation. The POST2-based reconstruction uses accelerometer
measurements to adjust an estimated state using a POST2-based simulation developed
prior to entry to support EDL analyses and design. Results from both approaches are
discussed in this paper.
The ESA Huygens probe science teams provided several sets of data. These data
included acceleration measurements throughout EDL, pressure and temperature after
heatshield jettison, and radar altimetry for the last 40 kilometers. Additionally, radio
telescopes at Earth received the Huygens signal throughout the descent and provided an
estimate of wind velocity. The Descent Imager (DI) team also contributed an estimate of
vertical velocity based on sequentially captured images.
The main emphasis of the Huygens probe reconstruction is to evaluate the
simulation models used prior to probe entry at Titan with actual flight data. Another
objective of the Huygens reconstruction was to compare the NASA derived trajectory and
the Huygens DTWG solution to the trajectory profile.

RECONSTRUCTION APPROACHS

Traditional Method

Using the traditional method, the trajectory reconstruction of the Huygens probe
trajectory parameters from entry interface to the surface of Titian is separated into two
parts. First, the probe trajectory on the Drogue parachute (lasting about 8650 s) is
addressed. Next the probe entry phase trajectory (the main deceleration pulse), leading to
the parachute phase and culminating just prior to the Drogue deploy (lasting about 350 s)
is reconstructed.
The reconstruction of the trajectory during the Drogue phase of the Huygens
probe descent to the surface is obtained using the pressure, temperature, and acceleration
data. In essence, the method involves integrating a form of the hydrostatic equation to
obtain altitude, and applying the aerodynamic force equation to get total relative velocity
magnitude. This approach is taken to circumvent the problems associated with integrating
the accelerometer data during long time intervals at near terminal velocity conditions.
The second half of the reconstruction was done in a more classical sense. It
involved the hypersonic portion of the flight starting from entry interface. By using the
acceleration data along with the navigation team provided state at entry interface the
trajectory was reconstructed by integrating the accelerations. After the trajectory was
reconstructed the atmosphere could be backed out using the aerodynamic database.

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Atmosphere Reconstruction
The accelerometer data from an entry probe during descent is the principle data
used to obtain the free-stream density. The aerodynamic force equation along the x-body
axis can be expressed as,

2ma
!= x
v2C S
r A

where the probe mass, m, reference area, S, and the axial aerodynamic force coefficient,
C are known a priori, and v is the probe velocity relative to the atmosphere. Given the
A r
reconstruction trajectory state variables (i.e. probe position, velocity, and orientation as a
function of time); density mapping is relatively straight forward, except for the C , which
A
is also a function of density (along with other parameters). This transcendental nature of
the density equation above can be overcome by a simple iterative technique.
The aerodynamic coefficient for the hypersonic phase, C , is determined using the
A
preflight database created by NASA Langley. This database requires the current values
for the center-of-gravity location, Knudsen number, Mach number, relative velocity;
angle of attack and sideslip angle are assumed zero. During the parachute/drogue phase,
the composite C S coefficient is determined as a linear combination of probe and
A
parachute drag. That is, for the total C S term the product of the approximate drag
A
coefficient, C (.8) and the heat shield reference area (5.73), or the descent module
D
reference area (1.29) was added to the parachute drag (a function of Mach Number and
Reynolds Number) times its reference area.

Drogue-Phase Equations of Motion (Hydrostatic equation)
The equation for the change in pressure due to a change in altitude of a fluid, such
as an atmosphere, is known as the hydrostatic equation, and is given as:

dp
="g! (1)
dh

Using the chain rule, substituting in equation (1) and solving for the altitude time
derivative gives:

dh p&
= (2)
dt "g!

Putting the density (from the equation of state using the measured temperature and
pressure) and gravity into equation 2, gives a transcendental equation that is a function of
the measured quantities that can be evaluated and integrated to get the vertical velocity
component and the altitude time history.

dh R(R +h)2T!p& "
= o (3)
# $
dt %µC W &p’
f MM
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This equation is integrated using the Ordinary Differential Equation (ODE) solver
that is built into MatLab. It requires that the pressure, temperature, mean molecular
weight, and pressure derivative be given as a function of time. The gravity and
compression factor are functions of altitude but since an ODE solver is being used, which
uses the previous time point to calculate the derivative to get to the next point, the
altitude is known. Since the time of surface impact is known, the integration is done from
the surface up. In summary, from the measurements of pressure and temperature as a
function of time, two of the six trajectory state variables (h, h&) as a function of time are
obtained.

POST2-based Method

A six degree-of-freedom atmospheric entry and three degree-of-freedom
parachute descent trajectory of the Huygens probe was simulated in Program to Optimize
Simulated Trajectories II (POST2).11 POST2 is a generalized point mass, discrete
parameter targeting and optimization trajectory program. POST2 has the ability to
simulate three and six degree-of-freedom (3DOF and 6DOF) trajectories for multiple
vehicles in various flight regimes. POST2 also has the capability to include different
atmosphere, aerodynamics, gravity, propulsion, parachute and navigation system models.
Many of these models have been used to simulate the entry trajectories for previous
missions (i.e., MER, Genesis, Mars Pathfinder)15-17 as well as current and planned NASA
missions (Stardust, Mars Phoenix, and Mars Science Laboratory).18,19 A variety of
system studies have been conducted and their atmospheric trajectories simulated in
POST2 including aerocapture at Titan, Neptune and Venus.20
Exploiting the modular nature of the POST2 program by adding mission specific
models in concert with the existing POST2 architecture allows for the development of
higher fidelity, mission specific simulations. These simulations usually have integrated
mission-specific engineering models and flight software that have been tested and
validated prior to the actual mission. These simulations support design, development,
testing, and operations of vehicles for particular missions. These POST2-based mission
specific simulations have been used operationally for day-of-entry flight software
parameter determination and prediction of mission metrics (such as touchdown footprint)
as well as aerobraking orbit prediction and assessment. Simulation complexity varies
from first-order trades (e.g. parachute size and deployment conditions, terminal descent
engine size, etc.) to all-up Monte-Carlo simulations for flight operations.
POST2 was used to simulate the Huygens entry trajectory into Titan.10 The
POST2-based flight simulation incorporated several models specific to the Huygens
probe entry: a 6DOF aerodynamics model; Titan’s gravity and Titan-GRAM atmosphere
(including wind) models; attitude inputs and initial states; trigger criteria, inflation, and
drag models for the Pilot, Main, and Drogue parachutes; as well as vehicle geometric
parameters. Note that version 1.0 of the Titan-GRAM atmospheric model was
implemented into POST2, with updates from Cassini measurements of Titan (the T0 and
TA atmospheric profiles). This simulation was used to produce single trajectory data and
was an integral element of the Monte Carlo analyses discussed in Ref. 10.

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Extended Kalman Filter POST2 Module
An extended Kalman filter (EKF) module has been developed for and included
into POST2. This module was developed and integrated into the POST2 software to
facilitate trajectory reconstruction using POST2-based mission specific simulations. The
general nature of POST2 inputs was retained for this module. That is, the state to be
estimated can be input as any POST2 input and the observation can be any POST2
outputs. While the general POST2 software architecture was retained, separate files of
observations and their associated weightings versus simulation time are required for use
with this EKF module. The utility of integrating this EKF function into POST2 is to
allow more rapid setup and execution of trajectory reconstruction runs using the same
simulation that has been tested and validated for that particular mission.
The theory and equations defining this module are well described in the
literature.21,22 A summary of the method implemented for the POST2 module is as
follows:

1) Define initial covariance and estimated state (P and X )
0 0
2) Read observations (O) and their weights (W = R-1) from files
3) Integrate to the next observation time:
X˙ =F(X,t) and P˙ = A*P +P *AT +Q"K*R*KT

where P is the propagated state covariance matrix; A is the matrix of state derivative
"X˙
! with respect to time! ( X˙ ) sensitivity to the state ( ); Q is the system noise covariance
"X
matrix (where the random noise is chosen to apply to every state); matrix K is from the
!
last update; and R (the measurement noise covariance) is set to the inverse of the
observation !w eightings matrix (W).
!

4) At the observation time, calculate:
"G T T "1
Y = O - G H = K =P *H *(H*P *H "R)
"X

where O is the observation matrix (from file), G is the matrix of calculated observation
values from the simulation and K is the Ka!lm an gain.
!

5) Next update the state:
ˆ
X = X +K*Y

6) And update the covariance:

!
Pˆ =(I"K*H)*P *(I"K*H)T +K*R*KT

ˆ ˆ
where I is the identity matrix, X is the estimated state, and P is the updated covariance.
This process is continued with POST2 integrating the state and covariance between
!
observation updates.

! !
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Huygens EDL Measured Datasets

The Huygens entry, descent, and landing datasets described below were provided
to NASA as part of an agreement with ESA for analyses NASA performed prior to entry.
These datasets include measurements of atmospheric pressure, temperature, composition,
and zonal winds, vehicle acceleration, and vehicle altitude. Various Principal
Investigators were responsible for the instruments that provided this data; the various
groups responsible for this data are identified, along with details about their instruments,
in Refs. 1 and 2.

Pressure
The time history of the reduced pressure was obtained from ESA as described
above. Figure 2 (top panel) shows the free-stream pressure from the onboard pressure
transducer as a function of time where the reference time is entry interface. (During the
day of entry, Jan. 14, 2005, t = 09:05:52.523)
EI

Figure 2: Upper Panel: Free-stream pressure provided by ESA.
Lower Panel: Pressure derivative using a 51-point sliding window with a third
degree polynomial evaluated at the mid-point.

Derivative of Pressure
The time derivative of pressure is needed for the calculation of altitude, see
equation (3). The method used to obtain the pressure derivative includes fitting a third
degree polynomial (using least-squares) to a 51-point sliding window of pressure data.
That is, at any time point, 25 pressure data to the left and right are fitted to a 3rd-degree
polynomial from which a derivative in obtained for the time point. One point to the right
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is added, while one point to the left is dropped from the original data set and the process
is repeated. This method clearly leaves 25 points at the beginning and at the end of the
data set without a derivative. To handle these end points, the window size is reduced to 5
points, leaving only two points without a derivative. Figure 2 (lower panel) shows the
results of the calculations for the time derivative of pressure, dp/dt. The derivative is
relatively smooth except at the end data points, as expected. The left end of the graph has
a little extra noise providing a few points with an unreliable derivative. On the ground,
the derivative goes to zero and this “corner” impacts the last few derivatives, as seen in
the graph. Neither end produces difficulty in the integration process since the points can
be readily omitted.

Density
The density is obtained from the calibrated measurements of pressure and
temperature using the equation of state. In addition, the mean molecular weight and the
compression factor (also taken from data provided through ESA, but not shown here) are
used in the calculation of density. Figure 3 shows the density results for both with and
without (Cf=1) compression factor. As expected, the difference is less than 3.5% and the
larger differences occur at lower altitudes, corresponding to larger values of time, as
shown in Fig. 3.

Figure 3: Density time history with and without compression effects.

Altimeter
Two separate independent altimeters, labeled “A” and “B”, were onboard the
Huygens probe. Figure 4 shows the data for both of these instruments after the Principal
Investigator has applied calibration factors. The data produced by both altimeter
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instruments lay on top of one another, although the altimeter B data starts at a higher
altitude.

Figure 4: Top Panel: Altimeter measurements from A and B instruments.
Lower Panel: Derivative of altimeter data using a 101-point moving window.

Accelerometer
The axial acceleration data set from the Huygens Servo Accelerometer, used in
the reconstruction is shown in Fig. 5. Note that Piezo Accelerometer data in all three axes
were received, but evaluation of this dataset deemed it unusable (the axial component
only reached 100 m/s2, well below the values expected and returned from two
independent sources, and forcing the lateral measurements to be rejected as well).
Examination of the Servo axial accelerometer data before entry interface indicated that
there is a bias in the data of about 22.8 µ-g in the data. In Fig. 5 the top panel shows the
data from entry interface to near probe touchdown on the surface of Titan. The lower
panel shows an expanded view of the data. In this panel the results of a smoothing
process is shown along with the original data. Smoothing was done on data beyond 330 s,
or right after parachute inflation. The smoothing process consists of a median process for
a sliding window, similar to what was done for the calculation of dp/dt and dh/dt,
discussed earlier, except a median is obtained instead of a curve fit. That is, a window
size of 201 data points is selected and a median of the data set is determined. The time
and median value are recorded at the middle of the data and subsequently, a data point is
added to the right and one deleted to the left and the process is repeated. Again, because
of the moving window the first and last 100 points of data in the original data set remain
not smoothed. The first 100 points are used to merge with the original data set, while the
last 100 points are not smoothed, as can be seen in Fig. 5. The spike in the acceleration
data at approximately 1100 seconds corresponds to drogue deploy which represents real
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data and as such the peak must be preserved so as to not lose the information. Thus, in
this time interval smoothing was not attempted. Similarly, the hypersonic portion of the
acceleration time history before main parachute deploy, i.e. before approximately 330
seconds, is also not smoothed so as not to lose the peak acceleration data. In summary,
the blue line in Fig. 5 represents the acceleration data used in the calculations reported in
the next sections.

Figure 5: Upper Panel: Huygens axial accel from Entry Interface to Titan’s surface.
Lower Panel: Expanded view of axial acceleration showing smooth and actual accel data.

RESULTS

Traditional Method

Figures 6, 7, and 8 show the atmosphere results for the hypersonic reentry phase
of the mission (denoted “Hyper” in the legend), namely density, pressure, and
temperature as a function of altitude, respectively. Included on each graph is the Huygens
model atmosphere generated by Yelle and the Titan-GRAM Atmosphere used in
simulations. Also included in the figures is the lower atmosphere reconstructed density
using the equation of state (denoted “Hydro” in the legend) as well as the measured
pressures and temperatures.